Bilinear Dynamical Networks Under Malicious Attack: An Efficient Edge Protection Method

In large-scale networks, agents and links are often vulnerable to attacks. In this paper, we focus on continuous-time bilinear networks, where additive disturbances model attacks or uncertainties on agents/states (node disturbances), and multiplicative disturbances model attacks or uncertainties on couplings between agents/states (link disturbances). We then investigate network robustness notion in terms of the underlying digraph of a network, and the structure of exogenous uncertainties and attacks. Specifically, we define the robustness measure using the
$\mathcal{H}_{2}$
-norm of the network and calculate it in terms of the reachability Gramian of the bilinear system. The main result is that under certain conditions, the measure is supermodular over the set of all possible attacked links. The supermodularity property facilitates the efficient solution of the optimization problem. We conclude with a few examples illustrating how different structures can make the system more or less vulnerable to malicious attacks on links.